We pay $319/month for kindergarten. It is at a public school but we have to pay it until 1st grade. Is this deductible? We receive no form from the school.

It depends, but probably not.  If it was prekindergarten, that could be considered a day-care expense that can be claimed by the Child and Dependent Care Credit.  However, if it is regular kindergarten (even with a private school), there is neither a Child and Dependent Care Credit nor any type of Educational Credit or deduction available, unless there is after-school care involved.  Please see this FAQ for more information:  https://ttlc.intuit.com/replies/3301288

The FAQ mentions, however, that some states do provide a credit for Private Education on the state return..